Lucre Ti Us 3 Spring

8/8/2019 Lucre Ti Us 3 Spring

http://slidepdf.com/reader/full/lucre-ti-us-3-spring 1/34

On the Nature of Things



g



Void

Movement requires void = empty space.



Void What is the nature of space?

A lecture about ³nothing´²for 50 minutes!

Universe is boundless (I, 960-970) ± If it did have a boundary, boundary has 2

sides. What¶s on the other side?

@

Universe is infinite Center is what is equidistant from boundaries.

@ Universe has no center (I, 1050-1070)



Void

Universe is boundless (I, 960-970)

@ Universe is infinite.

Is this a valid argument? Inference works in ordinary

circumstances:



Void

Universe is boundless (I, 960-970)

@ Universe is infinite.

Is this a valid argument? Inference works in Euclidean space.



Euclid(325-265 BC)

³E

lements´Treatise on

Math &

Geometry



Euclid¶s Parallel Postulate:Through a point not on a given line, there is one

and only one line that goes through that point

that is parallel to the given line.

Lines are ³parallel´ if they never intersect.



Void

Any space that satisfies Euclid¶s Parallel

Postulate is a Euclidean space.

Lucretius¶ inference works in Euclideanspaces.

Are all spaces Euclidean?

No

Non-Euclidean spaces

Non-Euclidean Geometry.



Non-Euclidean Geometry

Nicholai LobachevskiRussian mathematician

(1793-1856)

First proposed

non-Euclidean

geometry



Non-Euclidean Geometry

Georg Riemann(1826-1866)

German mathematician

First to formalize

non-Euclidean

geometry.



Surface of

a sphere is a non-euclidean space.

³S

traight line´is the shortest

distance between

two points.

On a sphere that

is a ³great circle´

Equator & longitude

lines are examples.



LA to

Jerusalem

A segment

of a great

circle.



On the surf ace o f a spher e: Through a point

not on a given great circle, there is no great circle

that goes through that point that is parallel to(never intersects) the given great circle.

Great circle

Point not on thegiven great circle

All great circles through

that point will intersectthe equator somewhere.



Continental

U.S. isbounded &

finite«.

But, even

without

boundaries

surface is

f inite.



Non-E

uclideanS

pace The surface of a sphere is a non-

Euclidean space.

A non-Euclidean space can be

boundless, and yet finite.

Our universe is a non-Euclideanspace.



Albert Einstein(1879-1955)

Space is curved.



Space is curved

The surface of the earth looks flat overa small distance, but is curved.

Our space looks Euclidean over a smalldistance, but is curved.

If you shot an arrow that kept going, it

would eventually hit you in the back!

It would never hit a boundary, buttravel only a (long but) finite distance.



Space is curved

Lucretius¶ argument is invalid!

People who get outside the 2-dimensionalsurface of a sphere can see it is curved in

3 dimensions.

People who get outside our 3-dimensionalspace can see it is curved in 4 dimensions!



Space is curved

Positive curvature Negative curvature



Space is curved

The shortest way from one point to

another on a non-Euclidean 2-

dimensional surface is by leaving

that surface and entering another

dimension!



Shortest way

from here to

Hong Kong

is«

«throughThe Earth!



Space is curved

The shortest way from one point to

another in our non-Euclidean 3-

dimensional space is by leaving that

space and entering another

dimension!



³We are entering a hole in the space-time

continuum.´



Void Aristotle thought

the universe had a

center²the center

of the earth.

Everything in the

universe naturallymoved toward the

center.



Void Universe is boundless (I, 960-970)

Center is what is equidistant fromboundaries.

@ Universe has no center (I, 1050-1070)

Is that valid?

Surface of a sphere has a center, but it isnot in the surface of the sphere!

Universe could have a center, which isnot in the universe!



Void If no center, toward which things

naturally move, what accounts for

movement?

Weight of atoms causes them to naturallymove downward (II, 190).

But which direction is ³down´?

³Down´ only makes sense relative to a

given frame of reference²like left/right.

Democritus rejected special direction.



Void

Does space itself²the void²constitute a

frame of reference?

Is there a difference between a universewith just one atom at rest, and a

universe with just one atom moving at a

constant speed in a constant direction? Yes = Absolute Conception of Space.

No = Relative Conception of Space.



Isaac Newton

(1642-1727)

Absolute

conception of space.

Lucretius agrees.



Gottfried Leibniz

(1646-1716)Relative

conception of

space.

Democritus agrees.



Einstein¶s

Theory of Relativity.

Lucre Ti Us 3 Spring

Documents

Transcript of Lucre Ti Us 3 Spring